Years notation
Inspired by many people. Rules #Ordinals until \(\varepsilon_0\) are without affixes, but ∞^... can be written as 0 AE and any n AE can be written as ((n-1)AE)^^∞. #\(\omega\)'s year is a single ∞, which marks a limit year every ∞. #Years beyond ∞ are denoted X+Y where Y is the smaller expression. (\(\omega+1\) becomes year ∞+1) #Years can be multiplied (there is a ∞2, or even a ∞∞) by postfixing the multiplier. #Affixes can be iterated (thus things like 1 (AE)^9 or 1 AE AE exist. So there are things like 0(AE)^∞=0EEE, and 0(EEE(+^n))^∞=0 EEE(+^(n+1)). #For EEE, YOH and YOR (0 YOH is the 1st fixed point of EEE{}*+), (x (EEE(+^n))+1) (EEE(+^n-1))^∞)=x+1 EEE(+^n) and ((x EEE)+1)(AE)^∞ = (x+1) EEE. #For larger affixes the smaller afterwards doesn't make difference (1 EEE AE = 1 EEE < 1 AE EEE). #All EEEs/YOHs/YORs/COVs/SNDs/WNHs*/WFCs** are of form \(\#@\), where # is a suffix and @ is the operators placed in decreasing order (so EEE+* is illegal). #The 1st fixed point of X of Y is (X+1) of 0. Notation #\(X(O0)^n=X\Delta_0n\), supremum of delta-n is delta-(n+1) expressed in . #\(0\Delta_{0\Delta_{\cdots}1}1=0\alpha_0\) #\(\alpha\)'s supremum is \(\beta_0\), etc. and we diagonalize over this with \(\emptyset_x\) where x is the index of the function in the sequence (all functions are indexed 0) to call, after which we add acute accents to it. #For accented empty sets the shorthand is \(^{(n)}\)ø where n is expressed in . The supremum of this is called \(0\bar\emptyset\) (shorthand is \(^{(n,1)}\)ø. It has supremum of (n,2)ø, etc. #This has a supremum of \(1E_00\) and \(1E_0n\) is the (\(1+n\))th limit of empty set notation, and of \(1E_0\) the limit is \(2E_0n\) etc. #\((((...E_n0)E_n0)E_n0)E_n0=1E_{n+1}0\) (This is from old version of the notation.) #This has a supremum of \(1E_00\) and \(1E_0n\) is the (\(1+n\))th limit of empty set notation, and of \(1E_0\) the limit is \(2E_0n\) etc. #\((((...E_n0)E_n0)E_n0)E_n0=1E_{n+1}0\) (This is from the old version of the notation.) #The supremum of \(E_\alpha\) equals \(nEE_00\), of \(EE_\alpha\) \(nE_0^30\) (not to be confused with \(n\text{ EEE}@\),... #The supremum of E_^ (yes, I did mean super-sub-E notation, not ExE) (not \(E^\omega\), but iterated E notation power towers) equals 0 u. #The next u limits are the next limits of E_^, and limits of u are u+1 numbers, etc. #Continue with u+u as the shorthand of u+(1 u). #u's can be multiplied (i.e. uu=u^2 is supremum of u notation where the multiplier is too an u notation value) or exponentiated (i.e. u^u=u^^2 the supremum of u^n where n is an u notation value) etc. (like u{u}u={u,2,1,2} the supremum of u notation values) and even things like u&u, u^u&u exist. #The supremum of this (u^^u^^...&u) equals \(v_0\), of v0 the v1, etc. #We can continue this (of v (or equiv. w(0,x)) notation the limit is w(1,0); of w(1,x) w(2,0). When we reach the limit, cont. with w(1,0,0), of 3-arg. w w(1,0,0,0). We can extend this notation with w(x&y) where x is any expr., y is any number in any notation (incl. w()). #For even farther reaches we introduce Slashdot Notation (thereby SN): the supremum is 0/.1, the next 0/.2, etc., the limit of 0/.x (i.e. 0/.(0/.(...))) equals 1/.0. The next limits are 2/.0,... If we reach tve limit (i.e. (((...)/.0)/.0)) the next ordinal is 1//.0. Then comes ///., \(/^4.\), etc. Limit of the / iterator notation is 1/..0. Then come /..., \(/.^4\), etc. #After /., we add things like \(/_1._1\) (\(/_1.\) is supremum of SSN (simple SN)), and after this has reached the supremum, we continue with \(/_{0,1}.\), and we define an array notation (trailing 0s removed, sup. of previous elements are 1 added to the next). After the slash index reaches the sup., we continue the same with the dot. (Note: super- and subscripts can be mixed). #After the AISN (array indexed SN) reaches its limits, the next step is introducing Aleph notation (אN), with Aleph iterations (as seen in SN) and indices, after it things like \(1\aleph(0)_{@_0}\aleph(1)_{@_1}\cdots\) and indices are generalized to use matrices to go even farther (the results aren't iterated, the expressions get concatenated). Examples Ordinals *Phenolth Infinity: ∞ (AE)^Phenol *Phenol EEE Defs. *WNH = When Never Happens (is supremum of SND{}*+) **WFC = When Forever Ceases (is supremum of WNH{}*+) Category:Notations